Sequential fully implicit well model with tridiagonal matrix structure for reservoir simulation

ABSTRACT

A subsurface hydrocarbon reservoir is simulated by sequential solution of reservoir and well equations to simulate fluid flow inside the reservoir and well production rates. Sequential solution of reservoir and well equations treats wells as specified bottom hole pressure wells. This avoids solving large matrices resulting from the simultaneous solution of the reservoir and well equations which can be computationally very expensive for large number of unknowns and require special sparse matrix solvers. Such sequential solution involves regular reservoir system solvers complemented by small matrices for the numerical solution of the well bottom hole pressures. The solution is performed on tridiagonal matrices for the adjacent reservoir cells to the well cells at the perforated well intervals; and a vector of the unknown reservoir potentials for the adjacent reservoir cells.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation in part of, and claimspriority to, Applicant's co-pending, commonly owned U.S. patentapplication Ser. No. 16/037,070, filed Jul. 17, 2018; Ser. No.15/061,628, filed Mar. 4, 2016; Ser. No. 14/040,930, filed Sep. 30,2013, now U.S. Pat. No. 9,494,709, issued Nov. 30, 2016; and U.S. patentapplication Ser. No. 13/023,728, filed Feb. 9, 2011, now U.S. Pat. No.9,164,191, issued Oct. 20, 2015.

The present application is also related to commonly owned U.S. patentapplication Ser. No. ______, filed of even date herewith, entitled“Sequential Fully Implicit Well Model with Tridiagonal Matrix Structurefor Reservoir Simulation,” (Attorney Docket No. 0004159.051056) andhaving the same inventor as the present application.

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to computerized simulation of hydrocarbonreservoirs in the earth, and in particular to simulation of flowprofiles along wells in a reservoir.

2. Description of the Related Art

Well models have played an important role in numerical reservoirsimulation. Well models have been used to calculate oil, water and gasproduction rates from wells in an oil and gas reservoirs. If the wellproduction rate is known, they are used to calculate the flow profilealong the perforated interval of the well. With the increasingcapabilities for measuring flow rates along the perforated intervals ofa well, a proper numerical well model is necessary to compute thecorrect flow profile to match the measurements in a reservoir simulator.

It is well known that simple well models such as explicit or semiimplicit models could be adequate if all reservoir layers communicatedvertically for any vertical wells in a reservoir simulator. For thesemodels, well production rate is allocated to the perforations inproportion to the layer productivity indices (or total mobility).Therefore, the calculations are simple and computationally inexpensive.The structure of the resulting coefficient matrix for the reservoirunknowns remains unchanged. Specifically, the coefficient matrixmaintains a regular sparse structure. Therefore, any such sparse matrixsolver could be used to solve the linear system for the grid blockpressures and saturations for every time step for the entire reservoirsimulation model.

However, for highly heterogeneous reservoirs with some verticallynon-communicating layers, the above-mentioned well models cannot producethe correct physical solution. Instead, they produce incorrect flowprofiles and in some occasions cause simulator convergence problems.

With the increasing sophistication of reservoir models, the number ofvertical layers has come to be in the order of hundreds to representreservoir heterogeneity. Fully implicit, fully coupled well models withsimultaneous solution of reservoir and well equations have beennecessary to correctly simulate the flow profiles along the well andalso necessary for the numerical stability of the reservoir simulation.In order to solve the fully coupled system, generally well equations areeliminated first. This creates an unstructured coefficient matrix forthe reservoir unknowns to be solved. Solutions of this type of matricesrequire specialized solvers with specialized preconditioners. For wellswith many completions and many wells in a simulation model, this methodhas become computationally expensive in terms of processor time.

SUMMARY OF THE INVENTION

Briefly, the present provides a new and improved computer implementedmethod of solving well equations together with reservoir equations in areservoir simulation model having formation layers having vertical fluidflow and flow barrier layers with no vertical fluid flow. The computerimplemented method forms a reduced well model system for horizontal aswell as multiple vertical wells in a reservoir simulation model byassembling as a single flow layer the flow layers having flow betweenthe flow barriers in the reservoir model in the vicinity of the wellsonly. For a production rate specified well, the method then solves thereduced well model system by matrix computation (using a direct sparsematrix solver) for the bottom hole pressure of the well and reservoirunknowns for the grid blocks where well is going through (reduced wellmodel system). This method is repeated for the well or wells in areservoir simulator. Next, the method then solves the full reservoirsimulation model by treating each well as having the determined bottomhole pressure, and based on a steady state volume balance relationshipof the layer completion rates, formation pressures andtransmissibilities to determine completion rates for the layers of thewell or wells for the full reservoir model, and determine total rate forthe well from the determined completion rates for the layers of the wellor wells. The method then forms a record of the determined completionrates for the layers and the determined total rate for the well orwells. The simulator calculates the pressure distribution of multi-phaseflow fluid saturations distribution in the reservoir by using thecalculated perforation rates and the reservoir data assigned to thesimulator grid blocks. The simulator is thus able to calculate pressureand saturation distribution within the reservoir with given productionand injection rates at the wells at a given time.

The present invention provides a new and improved data processing systemfor forming a well model for horizontal as well as multiple verticalwells by reservoir simulation of a subsurface reservoir from a reservoirmodel having formation layers having fluid flow and flow barrier layerswith no fluid flow. The data processing system includes a processorwhich performs the steps of solving by matrix computation the reducedwell model system model for the bottom hole pressure of the well orwells and reservoir unknowns around the well or wells and solving thefull reservoir model by treating the well(s) as having the determinedbottom hole pressure, and based on a steady state volume balancerelationship of the layer completion rates, formation pressures andtransmissibilities to determine completion rates for the layers of thewell(s) for the full reservoir model. The processor also determinestotal rate for the well(s) from the determined completion rates for thelayers of the well(s). The data processing system also includes a memoryforming a record the determined completion rates for the layers and thedetermined total rate or rates.

The present invention further provides a new and improved data storagedevice having stored in a computer readable medium computer operableinstructions for causing a data processor in forming a well model forhorizontal as well as multiple vertical wells by reservoir simulation ofa subsurface reservoir from a reservoir model having formation layershaving fluid flow and flow barrier layers with no fluid flow to performsteps of solving by matrix computation the reduced well model systemmodel for the bottom hole pressure of the well or wells and reservoirunknowns and solving the full reservoir model by treating the well(s) ashaving the determined bottom hole pressure, and based on a steady statevolume balance relationship of the layer completion rates, formationpressures and transmissibilities to determine completion rates for thelayers of the well(s) for the full reservoir model. The instructionsstored in the data storage device also include instructions causing thedata processor to determine the total rate for the well(s) from thedetermined completion rates for the layers of the well or wells, andform a record the determined completion rates for the layers and thedetermined total rate for the well(s).

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B are schematic diagrams for a well model in a reservoirsimulator of multiple subsurface formation layers above and below a flowbarrier in a reservoir being formed into single layers according to thepresent invention.

FIGS. 2A and 2B are schematic diagrams for a well model in a reservoirsimulator of multiple subsurface formation layers above and belowseveral vertically spaced flow barriers in a reservoir being formed intosingle layers according to the present invention.

FIG. 3 is a schematic diagram of a well model for simulation based on anexplicit model methodology illustrated in radial geometry.

FIG. 4 is a schematic diagram of a well model for simulation based on afully implicit, fully coupled model methodology.

FIG. 5 is a schematic diagram of a well model for reservoir simulationwith a comparison of flow profiles obtained from the models of FIGS. 3and 4.

FIG. 6 is a schematic diagram of a finite difference grid system for thewell models of FIGS. 3 and 4.

FIGS. 7A and 7B are schematic diagrams illustrating the well modelreservoir layers for an unmodified conventional well layer model and awell layer model according to the present invention, respectively.

FIGS. 8A and 8B are schematic diagrams of flow profiles illustratingcomparisons between the models of FIGS. 7A and 7B, respectively.

FIG. 9 is a schematic diagram of a well layer model having two fracturelayers in radial coordinates.

FIG. 10 is a functional block diagram or flow chart of data processingsteps for a method and system for a fully implicit fully coupled wellmodel with simultaneous numerical solution for reservoir simulation.

FIG. 11 is a functional block diagram or flow chart of data processingsteps for a method and system for a sequential fully implicit well modelfor reservoir simulation according to the present invention.

FIG. 12 is a schematic diagram of a linear system of equations with atridiagonal coefficient matrix for an explicit well model for a onedimensional reservoir simulator with a vertical well

FIG. 13 is a schematic diagram of a linear system of equations with atridiagonal coefficient matrix for a fully implicit well model for a onedimensional reservoir simulator with a vertical well

FIG. 14 is a schematic diagram of a linear system of equations with atridiagonal coefficient matrix for a constant bottom hole pressure wellmodel for a one dimensional reservoir simulator with a vertical well forprocessing according to the present invention.

FIG. 15 is a functional block diagram or flow chart of stepsillustrating the analytical methodology for reservoir simulationaccording to the present invention.

FIG. 16 is a schematic diagram of a computer network for a sequentialfully implicit well model for reservoir simulation according to thepresent invention.

FIG. 17 is a schematic diagram of a tridiagonal coefficient matrix.

FIG. 18 is a schematic diagram of a linear system of equations with atridiagonal coefficient matrix.

FIG. 19 is a schematic diagram of a finite difference grid system for ahorizontal well model oriented in a y-axis direction according to thepresent invention.

FIGS. 20A and 20B are schematic diagrams of horizontal well models in areservoir simulator of multiple subsurface formation layers in the areaof a flow barrier in a reservoir, before and after being formed into areduced horizontal well model according to the present invention.

FIG. 20C is a schematic diagram of a linear system of equations for thereduced horizontal well model of FIG. 20B.

FIGS. 21 and 22 are functional block diagrams or flow charts of dataprocessing steps for a method and system for a reduced horizontal wellmodel according to the present invention.

FIG. 23 is a schematic diagram of a finite difference grid system for areservoir model with multiple wells according to the present invention.

FIGS. 24A and 24B are schematic diagrams illustrating notations for gridnomenclature in the multiple well reservoir model of FIGS. 19 and 23.

FIGS. 25A, 25B, 25C and 25D are schematic diagrams illustrating gridnumbering for a three-dimensional reservoir model.

FIG. 26 is a functional block diagram or flow chart of data processingsteps for a method and system for a modification of the block diagram ofFIG. 11 for multiple wells in a reservoir with a fully implicit fullycoupled well model with simultaneous numerical solution for reservoirsimulation.

FIG. 27 is a schematic diagram of a linear system of equations for asimplified two well, three-dimensional reservoir model.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

By way of introduction, the present invention provides a sequentialfully implicit well model for reservoir simulation. Reservoir simulationis a mathematical modeling science for reservoir engineering. The fluidflow inside the oil or gas reservoir (porous media) is described by aset of partial differential equations. These equations describe thepressure (energy) distribution, oil, water and gas velocitydistribution, fractional volumes (saturations) of oil, water, gas at anypoint in reservoir at any time during the life of the reservoir whichproduces oil, gas and water. Fluid flow inside the reservoir isdescribed by tracing the movement of the component of the mixture.Amounts of components such as methane, ethane, CO₂, nitrogen, H₂S andwater are expressed either in mass unit or moles.

Since these equations and associated thermodynamic and physical lawsdescribing the fluid flow are complicated, they can only be solved ondigital computers to obtain pressure distribution, velocity distributionand fluid saturation or the amount of component mass or moledistribution within the reservoir at any time at any point. This onlycan be done by solving these equations numerically, not analytically.Numerical solution requires that the reservoir be subdivided intocomputational elements (cells or grid blocks) in the area and verticaldirection (x, y, z—three dimensional spaces) and time is sub-dividedinto intervals of days or months. For each element, the unknowns(pressure, velocity, volume fractions, etc.) are determined by solvingthe complicated mathematical equations.

In fact, a reservoir simulator model can be considered the collection ofrectangular prisms (like bricks in the walls of a building). The changesin the pressure and velocity fields take place due to oil, water and gasproduction at the wells distributed within the reservoir. Simulation iscarried out over time (t). Generally, the production or injection rateof each well is known during the production history of a reservoir.However, since the wells go through several reservoir layers (elements),the contribution of each reservoir element (well perforation) to theproduction is calculated by different methods. This invention deals withthe calculation of how much each well perforation contributes to thetotal well production. Since these calculations can be expensive andvery important boundary conditions for the simulator, the proposedmethod suggests a practical method to calculate correctly the flowprofiles along a well trajectory. As will be described, it can be shownthat some other methods used will result in incorrect flow profileswhich cause problems in obtaining the correct numerical solution and canbe very expensive computationally.

Nomenclature

-   -   Δx, Δy, Δz=Grid dimension in x, y and z directions    -   k_(x), k_(y), k_(z)=permeability in x, y and z directions    -   p=pressure    -   ρ=Fluid (oil) density    -   g=gravitational constant    -   z=vertical depth from a datum depth    -   r_(o)=Peaceman's radius=0.2 Δx    -   r_(w)=wellbore radius    -   T_(x), T_(y), T_(z)=Rock Transmissibility in the x, y and z        directions

The equations describing a general reservoir simulation model andindicating the well terms which are of interest in connection with thepresent invention are set forth below in Equation (1):

$\begin{matrix}{{{{\Delta_{x}{\sum\limits_{j = 1}^{n_{p}}{T_{x}\rho_{ij}\lambda_{j}\Delta \; \Phi_{j}}}} + {\Delta_{y}{\sum\limits_{j = 1}^{n_{p}}{T_{y}\rho_{ij}\lambda_{j}\Delta \; \Phi_{j}}}} + {\Delta_{z}{\sum\limits_{j = 1}^{n_{p}}{T_{z}\rho_{ij}\lambda_{j}\Delta \; \Phi_{j}}}} + {\sum\limits_{j = 1}^{n_{p}}q_{ij}}} = \frac{\Delta_{t}n_{i}}{\Delta \; t}}{{i = 1},{\ldots \mspace{14mu} n_{c}\mspace{14mu} \ldots}}} & (1)\end{matrix}$

where Δx is the difference operator in the x direction, T_(x), T_(y),T_(z) are the rock transmissibilities in the x, y and z directions asdefined in Equation below), j stands for the number of fluid phases,n_(p) is the total number of fluid phases, usually 3 (oil, water andgas), Σ is the summation term, ρ_(i,j) is the density of component i inthe fluid phase j, λ_(j) is the mobility of the phase j (Equation 6),Φ_(j) is the fluid potential (datum corrected pressure) of fluid phasej, similarly Δ_(y) is the difference operator in y direction, and Δ_(z)is the difference operator in z direction of the reservoir, q_(i,w,k) isthe well term (source or sink) for the component i for grid block (cell)k, Δ_(t) is the difference operator in time domain, n_(i) is the totalnumber of moles for component i, and n_(c) is the total number ofcomponents in the fluid system (methane, ethane, propane, CO₂, etc.).

Equation (1) is a set of coupled nonlinear partial differentialequations describing the fluid flow in the reservoir. In the above setof equations n_(i) represents the ith component of the fluids. n_(c) isthe total number of components of the hydrocarbons and water flowing inthe reservoir. Here a component means such as methane, ethane, propane,CO₂, H₂S, water, etc. The number of components depends on thehydrocarbon water system available for the reservoir of interest.Typically, the number of components can change from 3 to 10. Equation(1) combines the continuity equations and momentum equations.

In Equation (1) q_(i,w,k) is the well perforation rate at locationx_(k), y_(k), z_(k) for component i and k is the grid block (cell)number. Again, the calculation of this term from the specifiedproduction rates at the well head is the subject of the presentinvention.

In addition to the differential equations in Equation (1), pore volumeconstraint at any point (element) in the reservoir must be satisfied:

$\begin{matrix}{{V_{p}\left( {P\left( {x,y,z} \right)} \right)} = {\sum\limits_{j = 1}^{n_{p}}\frac{N_{j}}{\rho_{j}}}} & (2)\end{matrix}$

where V_(p) is the grid block pore volume, P (x, y, z) is the fluidpressure at point x, y, z, N_(j) is the total number of moles in fluidphase j, ρ_(i) is the density of fluid phase j.

There are n_(c)+1 equations in Equations (1) and (2), and n_(c)+1unknowns. These equations are solved simultaneously with thermodynamicsphase constraints for each component i by Equation (3):

f _(i) ^(V)(n _(iV) ;n ₁ ,n ₂ ; . . . n _(c) ;P,T)=f _(i) ^(L)(n _(iL);n ₁ ,n ₂ ; . . . n _(c) ,P,T)  (3)

where f_(i) is the component fugacity, superscript V stands for thevapor phase, L stands for the liquid phase, n_(i) is the total number ofmoles of component i, P is the pressure and T represents thetemperature.

In the fluid system in a reservoir typically there are three fluidphases: oil phase, gas phase and water phase. Each fluid phase maycontain different amounts of components described above based on thereservoir pressure and temperature. The fluid phases are described bythe symbol j. The symbol j has the maximum value 3 (oil, water and gasphases). The symbol n_(p) is the maximum number of phases (sometimes itcould be 1 (oil); 2 (oil and gas or oil and water); or 3 (oil, water andgas)). The number of phases n_(p) varies based on reservoir pressure (P)and temperature (T). The symbol n₁ is the number of moles of component iin the fluid system. The symbol n_(c) is the maximum number ofcomponents in the fluid system. The number of phases and fraction ofeach component in each phase n_(i,j) as well as the phase density ρ_(j)and ρ_(i,j) are determined from Equation (3). In Equation (3), V standsfor the vapor (gas) phase and L stands for liquid phase (oil or water).

Total number of moles in a fluid phase j is defined by:

N _(j)=Σ_(i=1) ^(n) ^(c) n _(i,j)  (4)

Total component moles are defined by Equation (5).

N _(i)=Σ_(i=1) ^(n) ^(p) n _(i,j)  (5)

Phase mobility in Equation (1), the relation between the phases,definition of fluid potential and differentiation symbols are defined inEquations (6) through (9).

λ_(j) =k _(r,j)/μ_(j)  (6)

In Equation (6), the numerator defines the phase relative permeabilityand the denominator is the phase viscosity.

The capillary pressure between the phases are defined by Equation (7)with respect to the phase pressures:

P _(c)(s _(j) ,s _(j′))=P _(j) =P _({tilde over (j)}′)  (7)

The fluid potential for phase j is defined by:

Φ_(j) =P _(j) −gρ _(j) z  (8)

Discrete differentiation operators in the x, y, and z directions aredefined by:

$\begin{matrix}{{\Delta_{x} = \frac{\delta}{\delta \; x}},{\Delta_{y} = \frac{\delta}{\delta \; y}},{\Delta_{z} = \frac{\delta}{\delta \; z}},{\Delta_{t} = \frac{\delta}{\delta \; t}}} & (9)\end{matrix}$

where Δ defines the discrete difference symbol and U is any variable.

Equations (1), together with the constraints and definitions inEquations (3) through (9), are written for every grid block (cell) in areservoir simulator using control volume finite difference method (someof the grid blocks may include wells). Resulting equations are solvedsimultaneously. This is done to find the distribution of n_(i) (x, y, x,and t), P (x, y, z, and t) for the given well production rates q_(T) foreach well from which the component rates in Equation (1) are calculatedaccording to the present invention using the new well model formulation.In order to solve Equations (1) and (2), reservoir boundaries in (x, y,z) space, rock property distribution K (x, y, z), rock porositydistribution and fluid properties and saturation dependent data isentered into simulation.

According to the present invention and as will be described below, areduced well model system is formed which yields the same determinationof a calculated bottom hole pressure as complex, computationally timeconsuming prior fully coupled well models.

According to the present invention, it has been determined that for thegrid blocks where the well trajectory is going through where a number offormation layers communicate vertically, the communicating layers can becombined for processing into a single layer, as indicated schematicallyin FIGS. 1 and 2 for the well model. This is done by identifying thevertical flow barriers in the reservoir for the well cells, andcombining the layers above and below the various flow barriers of thewell cells. Therefore, the full well model system is reduced to asmaller dimensional well model system with many fewer layers forincorporation into a well model for processing.

As shown in FIG. 1, a well model L represents in simplified schematicform a complex subsurface reservoir grid blocks (cells) where the wellis going through which is composed of seven individual formation layers10, each of which is in flow communication vertically with adjacentlayers 10. The model L includes another group of ten formation layersaround the well 12, each of which is in flow communication verticallywith adjacent layers 12. The groups of formation layers 10 and 12 inflow communication with other similar adjacent layers in model L areseparated as indicated in FIG. 1 by a fluid impermeable barrier layer 14which is a barrier to vertical fluid flow.

According to the present invention, the well model L is reduced forprocessing purposes to a reduced or simplified well model R (FIG. 1A) bylumping together or combining, for the purposes of determining potentialΦ and completion rates, the layers 10 of the well model L above the flowbarrier 14 into a composite layer 10 a in the reduced model R.Similarly, the layers 12 of the well model L below the flow barrier 14are combined into a composite layer 12 a in the reduced well model R.

Similarly, as indicated by FIG. 2, a reservoir model L−1 is composed offive upper individual formation layers 20, each of which is in flowcommunication vertically with adjacent layers 20. The model L−1 includesanother group of seven formation layers 22, each of which is in flowcommunication vertically with adjacent layers 22. The groups offormation layers 20 and 22 in flow communication with other similaradjacent layers in model L−1 are separated as indicated in FIG. 2 by afluid impermeable barrier layer 24 which is a barrier to vertical fluidflow. Another group of nine formation layers 25 in flow communicationwith each other are separated from the layers 22 below a fluid barrierlayer 26 which is a vertical fluid flow barrier as indicated in themodel L−1. A final lower group of ten formation layers 27 in flowcommunication with each other are located below a fluid flow barrierlayer 28 in the model L−1.

According to the present invention, the well model L−1 is reduced forprocessing purposes to a reduced or simplified model R−1 (FIG. 2A) bylumping together or combining, for the purposes of determining welllayer potential Φ and completion rates, the layers 20 of the model L−1above the flow barrier 24 into a composite layer 20 a in the reducedmodel R. Similarly, other layers 22, 25, and 27 of the model L−1 belowflow barriers 24, 26 and 28 are combined into composite layers 22 a, 25a, and 27 a in the reduced model R−1.

The reduced well model systems or well models according to the presentinvention are solved for the reservoir unknowns and the well bottom holepressure. Next, the wells in the full reservoir simulation model systemare treated as specified bottom hole well pressure and solved implicitlyfor the reservoir unknowns. The diagonal elements of the coefficientmatrix and the right hand side vector for the reservoir model are theonly components which are modified in processing according to thepresent invention, and these are only slightly modified. A regularsparse solver technique or methodology is then used to solve for thereservoir unknowns. The perforation rates are computed by using thereservoir unknowns (pressures and saturations). These rates are thensummed up to calculate the total well rate. The error between thedetermined total well rate according to the present invention and theinput well rate will diminish with the simulator's non-linear Newtoniterations for every time step.

The flow rates calculated according to the present invention alsoconverge with the flow rates calculated by the fully coupledsimultaneous solution for the entire reservoir simulation modelincluding many wells. Because the present invention requires solving asmall well system model, the computational cost is less. It has beenfound that the methodology of the present invention converges if thereduced well system is constructed properly, by using upscaling properlywhen combining communicating layers.

It is well known that simple vertical well models such as explicit orsemi-implicit models have been generally adequate if all reservoirlayers communicate vertically. As shown in FIG. 3, an explicit wellmodel E is composed of a number Nz of reservoir layers 30 in verticalflow communication, each layer having a permeability k_(x,i) (here irepresents the layer number, not a component) and a thickness Δz_(i) anda perforation layer rate q_(i) defined as indicated in FIG. 3. The totalproduction rate q_(T) for the explicit model E is then the sum of theindividual production rates q_(i) for the Nz layers of the explicitmodel as indicated in Equation (3) in the same Figure.

For explicit models, the well production rate is allocated to theperforations in proportion to the layer productivity indices (or totalmobility). Therefore, the calculations are simple. The resultingcoefficient matrix for the unknowns remains unchanged, i.e., maintains aregular sparse structure, as shown in matrix format in FIG. 12.Therefore, any sparse matrix solver can be used to solve the linearsystem for the grid block pressures and saturations for every time step.

Well Models

The methodology of several vertical well models of a reservoir simulatoris presented based also for simplicity on a simple fluid system in theform of flow of a slightly compressible single phase oil flow inside thereservoir. However, it should be understood that the present inventiongeneral in applicability to reservoirs, and can be used for any numberof wells and fluid phases in a regular reservoir simulation model.

FIG. 6 illustrates the finite difference grid G used in this descriptionfor a vertical well model. As seen, a well is located at the center ofthe central cell in vertical directions. The models set forth below alsocontemplate that the well is completed in the vertical Nz directions,and the potentials in the adjacent cells:

Φ_(BW),Φ_(BE),Φ_(BN),Φ_(BS)

are constants and known from the simulation run (previous time step oriteration value). Here subscript B refers to “Boundary”, W indicateswest neighbor, E stands for East neighbor, and N and S stand for northand south respectively. Again Φ describes the fluid potential (datumcorrected pressure).

As can be seen in FIG. 6, the well is penetrating a number of reservoirlayers in the vertical direction represented by the index I, for i=1, 2,3 . . . Nz, with Nz is being the total number of vertical layers in thereservoir model. For each layer i, there are four neighboring cells atthe same areal plane (x, y). These neighboring cells are in theEast-West (x direction) and North-South (y direction).

The potential Φ of the East, West, North and South neighbors are knownfrom the simulator time step calculations, and those potentials are setin that they are assumed not to change over the simulator time step. Thevertical potential variations at the well location are then considered,but neighbor potentials which can vary in the vertical direction but areassumed to be known.

The steady state volume balance equation for the cell (i) in FIG. 6 isas follows:

T _(wi)(Φ_(Bw)−Φ_(i))+T _(Ei)(ΦBE−Φ _(i))+T _(Ni)(ΦBN−Φ _(i))+T_(Si)(Φ_(BS)−Φ_(i))+T _(Up,i)(Φ_(i−1)−Φ_(i))+T _(Down,i)(Φ_(i+1)−Φ_(i)−q _(i)=0  (10)

In Equation (10), T represents the transmissibility between the cells.The subscripts W, E, N, and S denote west, east, south and northdirections, and (i) represent the cell index.

The transmissibilities between cells for three directions are defined byEquation (11) below:

$\begin{matrix}{{T_{wi} = {k_{x,{i - {1/2}}}\frac{\Delta \; y_{i}\Delta \; z_{i}}{0.5\left( {{\Delta \; x_{i - 1}} + {\Delta \; x_{i}}} \right)}}}{T_{E,i} = {k_{x,{i + \frac{1}{2}}}\frac{\Delta \; y_{i}\Delta \; z_{i}}{0.5\left( {{\Delta \; x_{i + 1}} + {\Delta \; x_{i}}} \right)}}}{T_{Ni} = {k_{y,{i - {1/2}}}\frac{\Delta \; x_{i}\Delta \; z_{i}}{0.5\left( {{\Delta \; y_{i - 1}} + {\Delta \; y_{i}}} \right)}}}{T_{S,i} = {k_{y,{i + \frac{1}{2}}}\frac{\Delta \; x_{i}\Delta \; z_{i}}{0.5\left( {{\Delta \; y_{i + 1}} + {\Delta \; y_{i}}} \right)}}}{T_{{Up},i} = {k_{z,{i - {1/2}}}\frac{\Delta \; x_{i}\Delta \; y_{i}}{0.5\left( {{\Delta \; z_{i - 1}} + {\Delta \; z_{i}}} \right)}}}{T_{{Down},i} = {k_{z,{i + {1/2}}}\frac{\Delta \; x_{i}\Delta \; y_{i}}{0.5\left( {{\Delta \; z_{i + 1}} + {\Delta \; z_{i}}} \right)}}}} & (11)\end{matrix}$

In the above Equations (11), Δx_(i) is the grid block size (cell size)in the x direction for the grid block (cell) number i. Similarly, Δy_(i)is the grid block size (cell size) in the y direction for the grid block(cell) number I, and Δz_(i) is the grid block size (cell size) or gridlayer thickness in the z direction for the grid block (cell) number i.K_(z,i-−1/2) is the vertical permeability at the interface of the cellsi and i−1. Similarly, K_(z,1+1/2) is the vertical permeability at theinterface of the cells i and i+1. As seen in FIG. 6, the cell i is atthe center, i+½ interface between the cell i and i+1. For conveniencethe subscript j is dropped while expressing East-West flow. Similarly(i, j−1) is the north neighbor of the central cell (i, j). Therefore,the notation (i, j−½) is the interface between the central cell andnorth neighbor in the y direction.

The same notation is carried out for the south neighbor: (i, j+½)denotes the interface between the central cell (i, j) and south neighbor(i, j+1), in the y direction. For convenience in the above equations,subscript i is dropped while expressing y (or j) direction. As isevident, transmissibilities in Equation (11) are defined in a similarmanner.

In FIG. 14, the diagonal terms {tilde over (T)}_(c,i) are defined byEquation (17a) below, and the transmissibilities between cells for threedirections are as defined in Equations 11, as set forth above.

The term {tilde over (b)}_(i) in the FIG. 14 (right hand side) areexpressed by Equation (17b) in the text. It is extracted here asfollows:

{tilde over (b)} _(i)=−(PI _(i)Φ_(W) +T _(w,i)Φ_(BW) +T _(w,i)Φ_(BE) +T_(w,i)Φ_(BN) +T _(w,i)Φ_(BS))

where i=1, 2, . . . Nz, with Nz again being the total number of grids inthe vertical directions, the number of vertical layers. In thisextraction of Equation (17b), PI_(i) the layer productivity index isdefined by Equation (17) and the Φ_(B) potential terms are the knownboundary potentials at boundaries of the neighbor cells to the west,east, north and south of the central cell. As is conventional, theterms, cell and grid block are used interchangeably.

Conventional well models can be generally classified in three groups:(a) the Explicit Well Model; (b) the Bottom Hole Pressure Specified WellModel; and the Fully Implicit Well Model (Aziz K, Settari A, PetroleumReservoir Simulation, Applied Science Publishers Ltd, London 1979). Fora better understanding of the present invention, a brief review of eachwell model is presented.

Explicit Well Model

For an explicit well model, the source term q_(i) in Equation (10) isdefined according to Equation (12) by:

$\begin{matrix}{q_{i} = {\frac{k_{x,i}\Delta \; z_{i}}{\sum\limits_{i = 1}^{i = {Nz}}{k_{x,i}\Delta \; z_{i}}}q_{T}}} & (12)\end{matrix}$

where q_(i) is the production rate for cell (grid block) i where thewell is going through and perforated.

Substituting Equation (12) into Equation (10) for cell i results in

T _(Upi)Φ_(i−1) +T _(C,i)Φ_(i) +T _(Down,i)Φ_(i+1) =b _(i)  (13)

where

T _(c,i)=−(T _(Up,i) +T _(down,i) +T _(Wi) +T _(Ei) +T _(Ni) +T_(Si))  (14a)

and

$\begin{matrix}{b_{i} = {{\frac{k_{x,i}\Delta \; z_{i}}{\sum\limits_{i = 1}^{i = {Nz}}{k_{x,i}\Delta \; z_{i}}}q_{T}} - \left( {{T_{Wi}\Phi_{BW}} + {T_{Ei}\Phi_{BE}} + {T_{Ni}\Phi_{BN}} + {T_{Si}\Phi_{BS}}} \right)}} & \left( {14b} \right)\end{matrix}$

Writing Equation (13) for all the cells i=1, Nz around the well for thewell cells only results in a linear system of equations with atridiagonal coefficient matrix of the type illustrated in FIG. 12, whichcan be written in matrix vector notation as below:

A _(RR){right arrow over (Φ)}_(R) ={right arrow over (b)} _(R)  (15)

In Equation (15), A_(RR) is a (Nz×Nz) tridiagonal matrix, and Φ_(R) andb_(R) are (Nz×1) vectors. Equation (15) is solved by computer processingfor the reservoir unknown potentials Φ_(R) grid blocks where well isgoing through by a tridiagonal linear system solver such as the Thomasalgorithm.

Tridiagonal Matrices and Systems

Tridiagonal Matrices are the matrices with only three diagonals in themiddle, with real or complex number as the entries in the diagonals.These diagonals are called “lower diagonal”, “central diagonal” and“upper diagonal”. The remaining elements or entries of a tridiagonalmatrix are zeros. For example, FIG. 17 illustrates a tridiagonal matrixA of 8×8 matrix dimensions (or order n=8). In the FIG. 17, elements ofA; the ith element of A, namely a₁, a₂, a₃, a_(i), i=1, 8, representsthe lower diagonal; b₁, b₂, b₃, b_(i), the central diagonal; and c₁, c₂,c₃, c_(i), the upper diagonal elements.

An example tridiagonal system of equations is illustrated in FIG. 18.Solution of a linear system of equations with a Tridiagonal Matrix suchas defined above and shown in FIG. 18 is easily solved by Gaussianelimination. An example of a solution of equations in this manner iscontained at:https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm.

Thus the solution for x_(i) as shown below and in FIG. 18 performed bysolving the matrix relationships of Equations (16a) through (16e) asfollows:

$\begin{matrix}{c_{i}^{\prime} = \left\{ \begin{matrix}{\frac{c_{i}}{b_{i}};} & {i = 1} \\{\frac{c_{i}}{b_{i} - {a_{i}c_{i - 1}^{\prime}}};} & {{i = 2},3,\ldots \mspace{14mu},{n - 1}}\end{matrix} \right.} & \left( {16a} \right) \\\begin{matrix}{{b_{i}^{\prime} = 1};} & {{i = 1},2,\ldots \mspace{14mu},n}\end{matrix} & \left( {16b} \right) \\{d_{i}^{\prime} = \left\{ \begin{matrix}{\frac{d_{i}}{b_{i}};} & {i = 1} \\{\frac{d_{i} - {a_{i}d_{i - 1}^{\prime}}}{b_{i} - {a_{i}c_{i - 1}^{\prime}}};} & {{i = 2},3,\ldots \mspace{14mu},{n.}}\end{matrix} \right.} & \left( {16c} \right) \\{x_{n} = d_{n}^{\prime}} & \left( {16d} \right) \\{{{x_{i} = {d_{i}^{\prime} - {c_{i}^{\prime}x_{i + 1}}}};{i = {n - 1}}},{n - 2},\ldots \mspace{14mu},1.} & \left( {16e} \right)\end{matrix}$

Bottomhole Pressure Specified Well Model

Φ_(w) is the uniform potential along the wellbore open to productionusing conventional techniques according to techniques explained in theliterature, such as the textbooks by Muskat “Physical Principles of OilProduction”, McGraw-Hill Book Co. (1949) and “The Flow of HomogeneousFluids Through Porous Media”, McGraw-Hill Book Co. (1937). For thepurposes of modeling in the present context, the friction pressure dropalong the well is considered negligible. Assuming that Φ_(w) is known(or specified), the oil rate from the perforation is calculated byEquation (17) as follows:

$\begin{matrix}{\begin{matrix}{q_{i} = {{PI}_{i}\left( {\Phi_{i} - \Phi_{W}} \right)}} \\{= {\frac{2\pi \; k_{x,i}\Delta \; z_{i}}{\ln \left( {r_{o,i}/r_{w}} \right)}\left( {\Phi_{i} - \Phi_{W}} \right)}}\end{matrix}\quad} & (17)\end{matrix}$

where PI_(i) is the layer productivity index, Φ_(w) is the specifiedbottom hole potential (datum corrected pressure), Φi is the reservoirgrid block pressure where well is going through for grid block (cell) i,r_(o,i) is called Peaceman's well block radius for grid block i definedas 0.2Δx, r_(w) is the radius of the well.

The variables in Equation (17) are explained in the Nomenclature sectionabove.

Substituting Equation (17) into Equation (10) and collecting the termsfor the cell i, for cell i the following result occurs:

T _(Upi)Φ_(i−1) +T _(C,i)Φ_(i) +T _(Down,i)Φ_(i+1W) =b _(i)  (18)

Let

T _(ci)=−(T _(Up,i) +T _(down,i) +T _(Wi) +T _(Ei) +T _(Ni) ++T _(Si)+PI _(i))  (18a)

b _(i)=−(PI _(i)Φ_(W) +T _(Wi)Φ_(BW) +T _(Ei)Φ_(BE) +T _(Ni)Φ_(BN) +T_(Si)Φ_(BS))  (18b)

Equation (18) when written for all the grid blocks i=1, Nz results in amatrix system illustrated in FIG. 14. The matrix of FIG. 14 for thebottom hole pressure specified well model can be seen to be similar tothe matrix of FIG. 12, and in a comparable manner Equation (18) issimilar to Equation (13). The bottom hole pressure specified well modelcan be easily solved by matrix computer processing with a tridiagonalequation solver methodology.

Fully Implicit Well Model

Total production rate q_(T) for a well according to a fully implicitwell model is specified according to Equation (19).

$\begin{matrix}{{q_{T} - {\sum\limits_{i = 1}^{i = {Nz}}{{PI}_{i}\left( {\Phi_{i} - \Phi_{W}} \right)}}} = 0.0} & (19)\end{matrix}$

The individual completion rate q_(i) is calculated by Equation (17). Forthe implicit well model, the wellbore potential Φ_(w) is assumedconstant throughout the well but it is unknown.

Substituting Equation (19) into Equation (10) and collecting the termsfor the cell i for cell (i) arrives at the following expression:

T _(Upi)Φ_(i−1) +T _(C,i)Φ_(i) +T _(Down,i)Φ_(i+1W) −PI _(i)Φ_(W) =b_(i)  (20)

Let

T _(ci)=−(T _(Up,i) +T _(down,i) +T _(Wi) +T _(Ei) +T _(Ni) +PI_(i))  (20A)

b _(i)=−(T _(Wi)Φ_(BW) +T _(Ei)Φ_(BE) +T _(Ni)Φ_(BN) +T_(Si)Φ_(BS))  (20B)

Writing Equation (20) for all cells yields a linear system of equationswith the form illustrated in FIG. 13 upper diagonal solid linerepresents T_(Up,i) as defined by Equation (11), and the lower diagonalsolid line describes the elements called T_(Down,i) described inEquation (11). The central term T_(C,I) is defined by Equation (20A) andright hand side b_(i) is defined by Equation (20B).

The linear system of the matrix (Equation 20) of FIG. 13 can berepresented in vector matrix notation as below:

$\begin{matrix}{{\begin{bmatrix}A_{RR} & A_{RW} \\A_{WR} & A_{WW}\end{bmatrix}\begin{bmatrix}{\overset{\rightarrow}{\Phi}}_{R} \\\Phi_{W}\end{bmatrix}} = \begin{bmatrix}{\overset{\rightarrow}{b}}_{R} \\b_{W}\end{bmatrix}} & (21)\end{matrix}$

In Equation (21), A_(RR) is a (Nz×Nz) tridiagonal matrix, A_(RW) is a(Nz×1) vector (reservoir PI'S), A_(WR) is a (1×Nz) vector (PI's) andA_(WW) is (1×1) scalar. For this example:

$A_{ww} = {\sum\limits_{i = 1}^{N_{z}}{PI}_{NZ}}$

Writing in algebraic form:

A _(RR){right arrow over (Φ)}_(R) +A _(Rw)Φ_(W) ={right arrow over (b)}_(R)  (22)

A _(wR){right arrow over (Φ)}_(R) +A _(ww)Φ_(W) =b _(w)  (23)

Solving for Φ_(W) from Eq 22 results in:

Φ_(W) =A _(ww) ⁻¹(b _(w) −A _(wR){right arrow over (Φ)}_(R))  (24)

Substituting into Equations (21)

A _(RR){right arrow over (Φ)}_(R) +A _(Rw)(A _(ww) ⁻¹(b _(w) −A_(wR){right arrow over (Φ)}_(R)))  (25)

(A _(RR) −A _(ww) ⁻¹ A _(wR) A _(RW)){right arrow over (Φ)}_(R) ={rightarrow over (b)} _(R) −A _(Rw) A _(ww) ⁻¹ b _(w)  (26)

Collecting the terms in Equation (25) produces:

The coefficient matrix (A_(RR)−A_(ww) ⁻¹A_(wR)A_(RW)) of Equation (26)is an (Nz×Nz) full matrix.

The resulting coefficient matrix can be defined as follows:

Ã=A _(RR) −A _(ww) ⁻¹ A _(wR) A _(RW)  (27)

and

{tilde over (b)}={right arrow over (b)} _(R) −A _(Rw) A _(ww) ⁻¹ b _(w)

Then Equation (26) can be written as:

Ã{right arrow over (Φ)} _(R) ={tilde over (b)} _(R)  (28)

The matrix of Equation (28) can be solved by a direct solver by Gaussianelimination or any other suitable conventional solver for the fullmatrices.

If in the implicit well model the number of layers Nz is large, andwells are fully completed in all the layers, then the solution ofEquation (28) becomes expensive in computation time. For this reason andalso if many wells are involved, generally Equation (28) is solved byiterative methods. An example of this is described in “A Fully-ImplicitFully-Coupled Well Model for Parallel Mega-Cell Reservoir Simulation”,SPE Technical Symposium of Saudi Arabia Section, 14-16 May 2005.

A flow chart I (FIG. 10) indicates the basic computer processingsequence of fully implicit, fully coupled well model simultaneoussolution for the type of matrix illustrated in FIG. 13. During a step100, simulation begins by reading the reservoir and production data.Reservoir data includes reservoir geometric information-its size, extent(length) in x, y, and z directions, reservoir properties such aspermeability distribution, porosity distribution, thickness of layers,relative permeability data, capillary pressure data, fluid property datasuch as fluid density tables, formation volume factor tables, viscositytables, location of wells, location of well perforations within thereservoir.

Production data includes oil, water, and gas production rate measured orspecified for the wells defined in the previous step. Production dataalso includes minimum bottomhole pressure for each well.

In many instances, only oil production rates and water production ratesare input if the gas production data is not available. If no water isproduced in the field, only oil production rates are read as input.

During step 102 the time step is incremented by one, and an iterationcounter of the number of non-linear iterations performed during thepresent time step is set to zero. During step 104, a Jacobian matrix ofreservoir data is formed. In step 106, the resultant linear system ofEquation (19) is then solved by iterative methods using sparsepreconditioners (Yousef Saad, Iterative Methods for Sparse LinearSystems, Society of Industrial & Applied Mathematics (SIAM) Publication,2003).

During step 108, a convergence step is performed to determine whetherthe non-linear iterations have converged. The individual residuals ofthe equations resulting from step 106 are checked against user-specifiedtolerances. If these tolerances are satisfied, the non-linear iterationloop is exited, the solution output is written to file for the currenttime step and processing returns to step 102 for the time step to beadvanced, and processing for the incremented time step then proceeds asindicated. If the user-specified tolerances are determined not satisfiedduring step 108, processing according to the non-linear iteration loopreturns to step 104 and continues. If the number on non-lineariterations becomes too large, a decision may be made to reduce the timestep size and go back to step 102.

However, based on the strength of the preconditioning, this method canalso be very expensive in computation time, since there is no exact wayof representing the full matrix in Equation (19). For difficultproblems, with extreme heterogeneity and small layer thicknesses, theiterative method may not converge.

Additionally, for highly heterogeneous reservoirs with some verticallynon-communicating layers, the above described well models do not producethe correct physical solution. Instead, they can produce incorrect flowprofiles, and in some occasions they can cause simulator convergenceproblems.

The Present Invention

Explicit and fully implicit models can produce totally different flowprofiles in case of some vertical flow barriers. This is displayed inFIG. 5. As shown in FIG. 5, a reservoir model V is composed of an upperlayer 50 of relatively low permeability, and with vertical flow, locatedabove an isolated high permeability layer 52 with no vertical flowcommunication with adjacent layers. The flow barrier layer 52 is locatedin the reservoir V above a layer 54 of medium permeability and havingvertical flow communication. As can be seen in FIG. 5, the productionrates qT for the well model V are the same for both implicit andexplicit well modeling methods.

However, the production rates q1, q2, and q3 of layers 50, 52 and 54differ significantly. The production rate q2 for the layer 52 is themajor contribution to the total production rate qT as shown by curve 56for the explicit model. On the contrary, for the implicit model as shownby curve 58, production rate q2 for the layer 52 is markedly less.

In the fully implicit well model, the production rate for the layer 2 issignificantly less due to the fact that this method takes into accountinternal reservoir heterogeneity (all the reservoir propertiesthroughout the reservoir) and heterogeneity around the well blocks inaddition to the perforation index (or layer 2 property alone as is usedby the explicit method as the only data to assign the rate fraction tothis perforation). For example, the fully implicit well method sees thatthere is no fluid supplied into layer 2 from layer 1 and layer 3 due tothe impermeable barrier between layer 2 and layer 1 and 3. Therefore,once some fluid is produced from well model layer 2, the pressure oflayer 2 should go down and this layer should not supply at a high rateinto the well, although the layer has very high permeability.

On the other hand, the explicit method assigns the rate for layer 2based on the permeability of this layer alone without considering thelayer connection to the above and below layers. Based on this, theexplicit method will assign a very high rate for this layer and willkeep it for the next simulation time step. In later simulation timesteps, this will cause instability in the production rates, i.e., thesimulator will reduce the time step size and will take a very long timeto complete the simulation. It is unsatisfactory for reservoirsimulation to have models provide divergent results for the same inputdata based on the modeling technique selected for use.

According to the present invention, a more comprehensive well model isprovided. The well model according to the present invention is called acoupled reservoir well model. The associated numerical solution isreferred to fully implicit, fully coupled and simultaneous solution. Afully implicit fully coupled reservoir well model produces correct flowprofile along the perforated well interval, as will be described. Asshown in FIG. 4 a reservoir model O is composed of a number z of iindividual layers 1 through Nz, each with a permeability k_(x,i) and athickness Δz_(i) and a potential Φ_(i) defined as indicated in FIG. 4,and upper and lower layers 40 of relatively low permeability, and withvertical flow, located above and below, respectively, an isolated highpermeability layer 42 with no vertical flow communication with adjacentlayers. As can be seen in FIG. 4, the production rates q, for layer i ofthe model O is an indicated by the expression set forth in FIG. 4.

The fully implicit coupled reservoir well model V of FIG. 5 is presentedin Equation (20) above and is also presented here in matrix format forfurther description:

$\begin{matrix}{{\begin{bmatrix}A_{RR} & A_{RW} \\A_{WR} & A_{WW}\end{bmatrix}\begin{bmatrix}{\overset{\rightarrow}{p}}_{R} \\{\overset{\rightarrow}{p}}_{W}\end{bmatrix}} = \begin{bmatrix}{\overset{\rightarrow}{b}}_{R} \\b_{W}\end{bmatrix}} & (21)\end{matrix}$

The present invention is based on the fact that the bottom hole pressureof a layered reservoir with a vertical well is the same as a systemaccording to the present invention. The system according to the presentinvention is formed by identifying flow barriers and lumping together orcombining for processing purposes the reservoir layers around the wellwhich are communicating in the vertical direction. Care should be takenaccording to the present invention in the formation of the reducedsystem. The reduced system must be formed correctly, or otherwise errorsin the reduced system construction can increase the total number ofnonlinear Newton iterations.

The system according to the present invention is solved for the bottomhole pressure. The solution is carried out by treating the well asspecified bottom hole pressure well. The procedure is fully implicit;however, it is not a simultaneous solution. Instead, the solution issequential. The method of the present invention is convergent since itis part of the global Newton iteration of the simulator. Therefore, ifthe model according to the present invention is constructed correctly,any possible error in rate calculations will be small and will diminishwith the simulator Newtonian iterations.

A flow chart F (FIG. 11) illustrates the basic computer processingsequence according to the present invention and the computationalmethodology taking place during a typical embodiment of a sequentialfully implicit well model for reservoir simulation with the presentinvention.

During a step 200, simulation begins by reading the reservoir andproduction data. Reservoir and production data read in during step 200are of the types discussed above. The reservoir simulator is alsoinitialized during step 200, setting the simulation day and the timestep to zero. During step 202 the time step is incremented by one, andan iteration counter of the number of non-linear iterations performedduring the present time step is set to zero.

During step 204, a Jacobian matrix of reservoir data is formed. In step206, a reduced system like that defined by the model R described aboveis formed according to the present invention and the matrix solved forbottom hole potential T in the manner described above. During step 208,the modified linear system matrix A is solved according to the presentinvention in the manner set forth above.

FIG. 15 illustrates the methodology of forming the reduced well modelsystem matrix R and solving for bottom hole potential Φ_(w) according tosteps 204 and 206 of FIG. 11. As indicated at step 210, vertical flowbarriers in the original well model system are identified. This can bedone based on well log data or by specification by a reservoir analystfrom data in the original reservoir model.

After steps 210, a reduced well model system model is then formed by thedata processing system D during a step 212. Those layers in the wellmodel which are located between flow barrier layers and have verticalflow are combined together for analytical model purposes.

Next, during step 214, the resulting reduced well model system is solvedby computer processing for bottom hole potential Φ_(w) and reservoirunknowns using the techniques discussed of Equations (17), (17a) and(17b). Step 216 then solves with the data processing system D the fullwell model system structural matrix of Equation (27) using a directsolver or other suitable technique described above. Completion rates q,and total well flow rate q_(T) are then determined with the dataprocessing system D based on the results of step 216.

Referring again to FIG. 11, during step 220, a convergence step isperformed to determine whether the non-linear iterations have converged.The individual residuals of the equations resulting from step 216 arechecked against user-specified tolerances. If these tolerances aresatisfied, the non-linear iteration loop is exited, the solution outputis written to file for the current time step and processing returns tostep 202 for the time step to be advanced, and processing for theincremented time step the proceeds as indicated. If the user-specifiedtolerances are determined not satisfied during step 208, processingaccording to the non-linear iteration loop returns to step 204 andcontinues. If the number on non-linear iterations becomes too large, adecision may be made to adjust the model.

Horizontal Wells

FIG. 19 displays a horizontal well 300 oriented in y direction locatedin a three dimensional reservoir model H. The model H of FIG. 19 is likeFIG. 6, a finite difference grid, but of horizontal well 300. The well300 is located in the center of a central cell in each of a successionof subsurface formation segments 304 in the y-direction extending invertical plans or in the z-direction. FIGS. 24A and 24B illustratenotations for grid cells in the horizontal wall model H of FIG. 19. Asseen in FIG. 19, the well 300 is completed in the horizontal ory-direction through N_(y) cells, and the potentials in the adjacentcells:

ΦB _(Up) ,ΦB _(E) ,ΦB _(Down) ,ΦB _(W)

are constants and known from the simulation run (previous time step oriteration value). The subscript B refers to boundary, Up indicates aboveneighbor, E indicates east neighbor, down indicates below neighbor, andW indicates west neighbor, with Φ again describing the fluid potentialor datum corrected pressure. In FIG. 19, the well 300 extendshorizontally along a longitudinal axis through each of a succession ofgrid blocks 302 in the y direction. If a known total well rate, q_(T),is input into a reservoir simulator, the reservoir simulator calculatesfor each time step the potential values Φ for each grid block 302 shownin FIG. 19.

As will be set forth, the present invention improves computerperformance as a reservoir simulator in forming measures of perforationrate qi for the layers which adds up to known total rate q_(T)

Similar to Equation (16), perforation rate q_(i) for each grid block canbe expressed as:

q _(i) =PI _(i)(Φ_(i)−Φ_(w))  (29)

In Equation (29), the productivity index is governed by

$\begin{matrix}{{PI}_{i} = \frac{2\pi \; k_{zy}\Delta \; y}{\ln \left( {r_{o,i}/r_{w}} \right)}} & (30)\end{matrix}$

The above expression of reservoir parameters and their physicalrelationships is according to Chen and Zhang, “Well Flow Models forVarious Numerical Methods”. International Journal of Numerical Analysisand Modeling, Vol 6, No 3, pages 378-388.

FIGS. 21 and 22 collectively represent a flow chart G which illustratesthe basic computer processing sequence according to the presentinvention and the computational methodology taking place during atypical embodiment of a sequential fully implicit well model forhorizontal well models, such as shown in FIG. 19 for reservoirsimulation with the present invention. During a step 400, simulationbegins by initializing the reservoir model H in the data processingsystem D and reading the reservoir and production data. The reservoirand production data read in during step 400 are of the types discussedabove for vertical well models in connection with the flow chart F andstep 200 (FIG. 11).

During step 400 an estimate of cell potentials Φ_(k) is formed for the kcells of the horizontal well model H. As indicated schematically in FIG.19, k=nx*ny*nz. During step 402, an estimate of well bore potential isformed according to Equation (32) as set forth below:

$\begin{matrix}{\Phi_{w} = {\frac{\sum{{PI}_{i}\Phi_{i}}}{\sum{PI}_{i}} - \frac{q_{t}}{\sum{PI}_{i}}}} & (32)\end{matrix}$

where Φ_(i) in Equation (32) is calculated from the initial potentialdistribution determined in step 400.

During step 404, perforation rates qi are determined for each cell Iaccording to the relationship expressed in Equation (29) above. In step406, the reservoir simulator is initialized and the simulator iterationcounter U set to zero. The simulation time step t is also initialized tozero for an initial iteration. The iteration counter and the time stepcounter are thereafter incremented, as will be set forth below, beforeperforming step 406 during subsequent iterations. The grid blockpotential for the initial time step is determined during step 406 bysolving a three dimensional potential equation with the reservoirsimulator according to Equation (1) for a single phase oil flow. Duringstep 408 boundary potentials are determined around each perforationΦ_(Bw), Φ_(B,Up), Φ_(BE), Φ_(B,Down) for each perforation (i), for i=1,2, Ny and stored in memory of the data processing system D.

Step 410 is then performed to form a reduced well model H−1 (FIG. 20B)by lumping the grid blocks 302 with no flow barriers among them as shownat 306 in FIG. 20A. For performing step 410, horizontal flow barrierssuch as shown at 303 in the original horizontal well model system H areidentified. This can be done based on well log data or by specificationby a reservoir analyst from data in the original reservoir model. Thereduced well model system model forming during step 410 results in thoselayers 302 in the well model H which are located between flow barriergrid blocks or layers 303 and have horizontal flow between them beingcombined together for analytical model purposes.

As can be seen in the example of FIGS. 20A and 20B, the horizontal wellmodel H of FIG. 20A with two groups of four grid blocks 302 on oppositesides of a flow barrier grid block 303 is transformed to reduced wellmodel H−1 as a result of step 410, the reduced well model H−1 having twogrid blocks 308, one on each side of flow barrier block 303.

The reduced well model equations for the reduced well model system H−1in FIG. 20B thus as shown in FIG. 20C becomes:

$\begin{matrix}{{\begin{pmatrix}T_{c\; 1} & T_{E\; 1} & 0 & {PI}_{1} \\T_{w\; 2} & T_{C\; 2} & T_{E\; 2} & {PI}_{2} \\0 & T_{w\; 3} & T_{C\; 3} & {PI}_{3} \\{PI}_{1} & {PI}_{2} & {PI}_{3} & {- {PI}_{T}}\end{pmatrix}\begin{pmatrix}{\overset{\sim}{\Phi}}_{1} \\{\overset{\sim}{\Phi}}_{2} \\{\overset{\sim}{\Phi}}_{3} \\\Phi_{W}\end{pmatrix}} = \begin{pmatrix}0 \\0 \\0 \\q_{t}\end{pmatrix}} & (18)\end{matrix}$

where PI_(T)=Σ_(i=1) ³PI_(i).

Then, during step 410 (FIG. 22) the reduced well model is solved for{tilde over (Φ)}₁, {tilde over (Φ)}₂, {tilde over (Φ)}₃, and {tilde over(Φ)}_(w), where {tilde over (Φ)} represents the grid block potential ofthe reduced horizontal well model H−1.

During step 412, the horizontal well model is converted to a fixedflowing bottom hole potential well using Φ_(w). Step 414 involvessolving the three diagonal matrix system of FIG. 20C for the reducedhorizontal well model H−1 shown in FIG. 20B to determine the potentialsfor each grid 302 shown in FIG. 19.

Referring again to FIG. 22, during step 416, a convergence step isperformed to determine whether the non-linear iterations have converged.The individual residuals of the equations resulting from step 414 arechecked against user-specified tolerances. If these tolerances aresatisfied, the non-linear iteration loop is exited, the solution outputis written to file and stored in memory of the data processing system Dfor the current time step and processing returns to step 418 for thetime step t to be advanced, and processing for the incremented time stepthe proceeds to step 406 as indicated. If the user-specified tolerancesare determined not satisfied during step 416, processing according tothe non-linear iteration loop returns to step 406 and continues. If thenumber on non-linear iterations becomes too large, a decision may bemade to adjust the model.

Multiple Vertical Wells

A 3 dimensional reservoir model M is shown in FIG. 23 with multiplevertical wells 500, each with assumed single phase, slightlycompressible oil flow. The model M is organized according to thenomenclature in the notations schematically illustrated in FIGS. 25A,25B, 25C and 25D.

The model M of FIG. 23 is, like FIG. 6 and, a finite difference grid,but of a number of vertical wells 500. The wells 500 are each located inthe center of a central cell in each of a succession of subsurfaceformation segments 504 in the z-direction extending in horizontally orin the y- and z-directions.

As in FIG. 6, each well 500 is completed in the vertical or z-directionthrough Nz cells, and the potentials in the adjacent cells:

ΦB _(N) ,ΦB _(E) ,ΦB _(S) ,ΦB _(W)

are constants and known from the simulation run (previous time step oriteration value). As in FIG. 6, the subscript B refers to boundary, Nindicates above neighbor, E indicates east neighbor, down indicatesbelow neighbor, and W indicates west neighbor, with 1 again describingthe fluid potential or datum corrected pressure.

Each well 500 extends vertically along a longitudinal axis through eachof a succession of grid blocks 502 in the z direction, and at certainformation layers at various depths each well 500 is intersected bycompletions 506. FIGS. 25A through 25D illustrate schematically thenumbering notations in the model M for wells 50 designated Well 1 andWell 2 in FIG. 23.

If a known total well rate, qT, for each well 500 is provided as aninput parameter into a reservoir simulator, the reservoir simulatorcalculates for each time step the potential values 1 for each grid block502 shown in FIG. 23.

For a generalized case of 3-dimensional vertical reservoir model M, thenumber of the wells is represented by n_(w), and the total well ratesfor each well 500 are given and denoted by q_(T)(l), l=1, n_(w) areknown from production data and provided as input parameters into thereservoir simulator.

FIG. 26 represent a flow chart for processing according to the presentinvention where multiple vertical wells are present, as is the case inthe model M of FIG. 23. Thus for the 3-dimensional vertical reservoirmodel M, in step 602, the reservoir simulator is initialized andreservoir and production are read from memory for processing. This isdone in a comparable manner to step 200 if FIG. 11. Simulationprocessing continues in 602 shown in FIG. 26, where an estimate of thewell bore potential Φw (l), l=1, 2, . . . nw is formed for each wellaccording to Equation (32), with the productivity index PI beingdetermined according to the measures expressed in Equation (17). Duringstep 604, the perforation rates q_(i) (l), l=1, . . . nw are computedfor each well, based on the estimates of well bore potentials resultingfrom step 602.

Step 606 involves forming of reduced well models for each well l=1, 2, .. . nw for the 3-dimensional multiple well model M. The reservoirsimulation is performed by combining or merging adjacent well cells 502of the formation layers which have fluid communication between eachother and are also located between flow barrier layers which have noflow through them. This is done in a manner for layers of a3-dimensional in a manner such as that shown schematically in FIGS. 1Aand 1B for layers adjacent a single vertical low well, and in FIGS. 20Aand 20B for layers adjacent horizontal flow wells.

In step 608, reduced well matrices are formed for each well based on thereduced well models form step 606, in a comparable manner to the reducedwell matrix of FIG. 20C for the horizontal well model H of FIG. 19.

Then, in step 610, bottom hole potential Φw (l), l=1, 2, . . . nw isdetermined for each well by solving the reduced well matrices resultingfrom step 608. In step 612, the diagonal {tilde over (T)}_(c,i) andright hand side b_(i) term of the main matrix for each well is modifiedaccording to the relationships according to Equations 18, 18A and 18Babove. FIG. 27 is an example schematic diagram of a reduced well matrixof a linear system of equations for a simplified two well,three-dimensional reservoir model, or 3×3×2 reservoir model using thenumbering system explained in FIGS. 24A and 24B and FIGS. 25A through25D.

During step 614, a total matrix, such as the example of FIG. 27, isformed for all the grid blocks 502 for the unknowns of the model M.Processing then after step 614 of FIG. 26 proceeds to convergencetesting in the manner of step 416 of FIG. 27 and iteration and time stepincrementing with return to step 602 for further processing iterationsfor the total system including each of the n_(w) wells.

Data Processing System

As illustrated in FIG. 16, a data processing system D according to thepresent invention includes a computer 240 having a processor 242 andmemory 244 coupled to the processor 242 to store operating instructions,control information and database records therein. The computer 240 may,if desired, be a portable digital processor, such as a personal computerin the form of a laptop computer, notebook computer or other suitableprogrammed or programmable digital data processing apparatus, such as adesktop computer. It should also be understood that the computer 240 maybe a multi-core processor with nodes such as those from IntelCorporation or Advanced Micro Devices (AMD), or a mainframe computer ofany conventional type of suitable processing capacity such as thoseavailable from International Business Machines (IBM) of Armonk, N.Y. orother source.

The computer 240 has a user interface 246 and an output display 248 fordisplaying output data or records of processing of well logging datameasurements performed according to the present invention to obtainmeasurements and form models of determined well production of formationlayers in a well or wells of subsurface formations. The output display48 includes components such as a printer and an output display screencapable of providing printed output information or visible displays inthe form of graphs, data sheets, graphical images, data plots and thelike as output records or images.

The user interface 246 of computer 240 also includes a suitable userinput device or input/output control unit 250 to provide a user accessto control or access information and database records and operate thecomputer 240. Data processing system D further includes a database 252stored in computer memory, which may be internal memory 244, or anexternal, networked, or non-networked memory as indicated at 254 in anassociated database server 256.

The data processing system D includes program code 260 stored innon-transitory memories 244 of the computer 240. The program code 260,according to the present invention is in the form of computer operableinstructions causing the data processor 242 to form a sequential fullyimplicit well model for reservoir simulation according to the presentinvention in the manner that has been set forth.

It should be noted that program code 260 may be in the form ofmicrocode, programs, routines, or symbolic computer operable languagesthat provide a specific set of ordered operations that control thefunctioning of the data processing system D and direct its operation.The instructions of program code 260 may be stored in memory 244 of thecomputer 240, or on computer diskette, magnetic tape, conventional harddisk drive, electronic read-only memory, optical storage device, orother appropriate data storage device having a computer usablenon-transitory medium stored thereon. Program code 260 may also becontained on a data storage device such as server 64 as a non-transitorycomputer readable medium, as shown.

Two illustrative example model problems are presented below: a sevenlayer homogeneous reservoir with one fracture flow barrier (FIG. 7); anda twenty-two layer heterogeneous reservoir with two fracture flowbarriers (FIG. 9).

Seven Layer Homogeneous Well Model

FIG. 7A illustrates the seven reservoir layers and properties for theoriginal model 70 and reduced model 71. As seen, it was assumed thatreservoir has seven layers. Layer thickness of each layer 72 withvertical flow is 10 ft. A layer 73 which represents a fracture with novertical flow is present which is 1 ft. thick. It was further assumedthat the layer 73 does not communicate with the layers 72 above andbelow. It was assumed that there is a vertical well in the middle, asindicated by an arrow. Initial reservoir potential (datum correctedpressure) is 3,000 psi for the model of FIG. 7A. Each layer 72 wasassumed to have 10 md areal permeability k_(x) and k_(y) and 1 mdvertical permeability k_(z).

Table 1 summarizes the reservoir and grid properties for the model 70.The grid size in the area direction (square grid) was assumed to be 840ft. Oil viscosity was set to 1 cp and Oil Formation Volume factor wasassumed to be 1. Well Total Oil Production rate was set to 1,000B/D. ALayer Productivity Index PI for each layer completion was calculated byPeaceman's method, as described, and is also shown in Table 1.

TABLE 1 Problem 1-Reservoir Properties Thickness, K_(x) = K_(y), K_(z),Layer, PI Layer ft md md b/d/psi 1 10. 10. 1 0.12 2 10. 10. 1 0.12 3 110,000 1.e−9 12.14 4 10 10 1 0.12 5 10 10 1 0.12 6 10 10 1 0.12 7 10 101 0.12

Fully Implicit Fully Coupled Simultaneous Solution

The coefficient matrix for the solution of reservoir pressures and thebottom hole pressure is formed in a similar manner explained above withregard to Equations (18-19) and as shown in FIG. 13. It can be seen thatthere are only 8 unknowns (7 potentials or datum corrected pressures)and the one bottom hole potential), and that the coefficient matrix isnon-sparse. The linear system of equations can be solved by a directmethod, such as Gaussian Elimination, for the unknown reservoir (layer)potentials Φ_(i), i=1, 7 and the other unknown Φ_(W).

Results

Table 2 summarizes the calculated layer potentials, wellbore potentialand layer (completion) flow rates for the model 70 of FIG. 7A.

TABLE 2 Exact Solution of the Original Problem Potential, Bottom hole(Wellbore) Layer psi Potential, psi Rate/d 1 2630.61 1257.36 166.67 22630.61 1257.36 166.67 3 1257.37 1257.36 0.0 4 2630.61 1257.36 166.67 52630.61 1257.36 166.67 6 2630.61 1257.36 166.67 7 2630.61 1257.36

From the computed results we see that computed bottom hole potential

Φ_(W)=1257.36 psi.

Formation of the Problem According to the Present Invention

According to the reservoir data in Table 1 and as shown in FIG. 7A,there is only one layer 73 which does not communicate vertically withthe other layers. Therefore, as shown in FIG. 7B, the layers 72 abovethe fracture layer 73 are combined into a single layer according to formthe reduced well model in accordance with the present invention.Similarly layers 72 below layer 73 are combined into a single layer. Thereduced model now can be seen to have only three layers. The totalnumber of the unknowns is 4 as opposed to 8 as in the full model.

Table 3 summarizes the properties of reduced well model 71 formedaccording to processing with the present invention.

TABLE 3 Reduced Well Model Layer Thickness, ft K_(x) = K_(y), md K_(z),md PI, b/d/psi 1 20 10 1 0.24 2 1 10,000 0 12.14 3 40 10 1 0.40

The linear system of equations (Equation 20) for the reduced systemstill has an unstructured coefficient matrix, but with a 50% less numberof unknowns. In actual reservoirs, with hundreds of layers and only afew flow barriers, the well model size reduction according to thepresent invention would be drastic, for example, a reduced well modelsystem model according to the present invention could be 1 percent ofsize of the full system. The reduced system is solved by a direct solverfor the layer potentials and the bottom hole potential. Table 4 presentsthe results.

TABLE 4 Results of the Reduced System Potential, Bottom hole (Wellbore)Layer psi Potential, psi 1 2630.61 1257.36 2 1257.37 1257.36 3 2630.611257.36

It can be seen that computed bottom hole potential:

Φ_(W)=1257.36 psi

is exactly the same as the Φ_(W) calculated for the full model.

The determined well potential is the only information needed for thenext step. The well is next treated as a specified bottom hole pressure(potential) model. Computer processing according to the proceduredescribed with the matrix of FIG. 14 and Equations (16 and 18) arefollowed to calculate the flow profile (layer rates) and the total wellrate. In FIG. 14, upper diagonal solid line of the matrix representsT_(Up,i) as defined by Equation (11), and the lower diagonal solid lineof the matrix describes the elements called T_(Down,i) as also definedby Equation (2). The central term T_(C,i) is defined by Equation (17a)and the right hand side bi defined by Equation (17b).

The results are summarized in Table 5. It is to be noted that totalcalculated well rate is exactly the same as input value of 1,000b/d.

TABLE 5 Results for the Total System with the Present Invention LayerPotential, psi Rate/d 1 2630.61 166.67 2 2630.61 166.67 3 1257.37 0.0 42630.61 166.67 5 2630.61 166.67 6 2630.61 166.67 7 2630.61 166.67 Total1,000.

Results presented in Table 5 are the same as in Table 1 for the fullyimplicit well model. Difference or error between well rates for thecalculated and input well is zero for this case and there no need for anextra iteration. This is because of the fact that the reservoir washomogeneous and no upscaling errors were made while forming the reducedsystem. Matrix diagonal elements and right hand side are the same as inFIG. 14, i.e., lower diagonal solid line represents T_(Up,i) defined byEquation (11), upper diagonal solid line describes the elements calledT_(Down,i) described above. The central term T_(C,i) defined by Equation(17a) and right hand side b_(i) defined by Equation (17b).

The terms PI appear on Equation (17) are the perforation productivityindexes for a square grid is defined by:

${PI}_{i} = {2\pi \; k_{x,i}\frac{\Delta \; z_{i}}{\ln \left( {0.2\Delta \; {x/r_{w}}} \right)}}$

where r_(w) is the well bore radius.

Comparison with the Explicit Well Model

In several reservoir simulators, semi implicit well models or explicitwell models are used. If the formulation of the well model issemi-implicit but it collapses to explicit in the pressure variable,this formulation collapses to explicit well models. The explicit wellmodel is for this problem obtained by following the computer processingprocedures for the matrix of FIG. 12 and Equations (12-14). In FIG. 12,the terms T_(Down,i), T_(Up,i) which appear on the diagonal elements aredefined by Equation (11), and T_(c,i), b_(i) are defined by Equation(14a) and Equation (14b). FIG. 8A illustrates the seven reservoir layersand properties for an implicit well model 80 and FIG. 8B an explicitwell model 81, of like structure to the model of FIGS. 7A and 7B. Asseen it was assumed that reservoir has seven layers. Layers 82 each havea potential Φ of 2630 psi. A layer 83 which represents a fracture andwas further assumed that does not communicate with the layers 72 aboveand below has a potential Φ of 1257 psi. It was assumed that there is avertical well in the middle, as indicated by an arrow. FIGS. 8A and 8Bcompare the results of Implicit and Explicit Models. Computedperforation (layer) rates are summarized in Table 6.

TABLE 6 Comparison of Perforation (Layer) Rates for Different WellModels Exact Solution New Explicit Layer/ (Fully Implicit Fully CoupledMethod Method Perforation Simultaneous Solution), Rate b/d Rate, b/dRate, b/d 1 166.67 166.67 9.43 2 166.67 166.67 9.73 3 0.0 0.0 943.40 4166.67 166.67 9.43 5 166.67 166.67 9.43 6 166.67 166.67 9.43 7 166.67166.67 9.43

As can be seen, the model 81 according to the explicit method isinaccurate; it totally miscalculates the perforation rate. The explicitmodel method assigns practically all the well production from the thinfracture layer 83 as indicated in FIG. 8B since this layer has thehighest productivity index.

The implicit methods (whether the computationally intensive fullyimplicit model or the reduced model according to the present invention)do not make such an assignment and instead determine that the layer 83is not getting fluid support from the layers 82 above and below. Theonly fluid support a fracture layer of this type shown at 83 whenpresent in an actual reservoir can get is from its planar neighboringcells. However, since the fracture layer is a very thin layer,transmissibility in these directions is by nature small. Therefore, thefracture layer cannot supply fluid at the rates simulated by theexplicit model.

In fact, conventional implicit well models show that during thetransient time the fracture layers support most of the well productionas do the explicit methods. However, the layer pressure in layer 83quickly declines and assumes the value of the uniform wellbore potential(constant bottom hole pressure). After the pressure declines, it reachessteady state, and the well production rate is in fact made by thecontribution from the layers 82 above and below the vertical flowbarrier 83.

Twenty-Two Layer Heterogeneous Reservoir Model

A model grid system 90 includes twenty two layers as shown in FIG. 9.The location of high permeability fracture layers 6 and 12 as countedmoving downward through the layer and indicated schematically at 91 and92. There are five layers 93, numbered 1 through 5 above layer 91, eachwith vertical flow. There are also five layers 94 in the model 90 withvertical flow between the flow barrier layers 91 and 92, and ten layers95 with vertical flow located below the flow barrier layer 92. Reservoirdata for the model 90 is shown in Table 7.

TABLE 7 Reservoir Data for 22 Layer Problem Permeability, VerticalThickness, mD Permeability, Layer ft (K_(x) = K_(y)) K_(z), mD  1 10 2 2 2 10 5 1  3 10 3 3  4 10 10 5  5 10 5 4  6 1 1,000. 1.e−9  7 10 6 6  810 3 3  9 10 9 6 10 10 12 2 11 10 5 5 12 1 1,000. 1.e−9 13 10 7.5 3.5 1410 7.5 3.5 15 10 7.5 3.5 16 10 7.5 3.5 17 10 7.5 3.5 18 10 9.2 1.2 19 109.2 1.2 20 10 9.2 1.2 21 10 9.2 1.2 22 10 9.2 1.2 Areal neighboring cellpermeabilities = 20 mD Total Well Production Rate = 2,500 B/D WellCompleted in all layers.

Results

-   -   Fully Implicit Fully Coupled Simultaneous Solution    -   Calculated bottom hole potential Φ_(W)=1421.247 psi

TABLE 8 Potential Distribution, psi Layer Φ_(W) Φ_(i)  1 1421.25 2901.32 2 1421.25 2900.82  3 1421.25 2900.38  4 1421.25 2900.38  5 1421.252900.38  6 1421.25 1421.25  7 1421.25 2864.43  8 1421.25 2864.37  91421.25 2864.10 10 1421.25 2863.88 11 1421.25 2864.03 12 1421.25 1421.2513 1421.25 2841.62 14 1421.25 2841.58 15 1421.25 2841.48 16 1421.252841.33 17 1421.25 2841.13 18 1421.25 2840.65 19 1421.25 2840.08 201421.25 2839.65 21 1421.25 2839.37 22 1421.25 2839.24

The model 90 was then subjected to the explicit model techniques of thetype described above and flow data determined. A comparison of flow ratedistribution for fully implicit and explicit processing of the reservoirmodel 90 using techniques previously described is set forth in Table 9.

TABLE 9 Comparison of Flow Rates for Fully Implicit Fully Coupled andExplicit Well Methods Layer Implicit Explicit  1 35.93 14.56  2 89.7836.39  3 53.85 21.83  4 179.48 72.78  5 89.74 36.39  6 0.00 727.80  7105.09 43.67  8 52.54 21.83  9 157.60 65.50 10 210.10 87.34 11 87.5536.39 12 0.00 727.80 13 129.29 54.59 14 129.28 54.59 15 129.28 54.59 16129.26 54.59 17 129.24 54.59 18 158.49 66.96 19 158.42 66.96 20 158.3766.96 21 158.34 66.96 22 158.33 66.96

Reduced Model Construction

Since there are only two vertical flow barriers layers 91 and 92, layers93 above layer 91 in FIG. 9 can be combined into one layer; layers 94below layer 91 into one layer, and layers 95 below layer 92 into anothersingle layer. Therefore the total number of layers according to thepresent invention is 5. The properties of the reduced model are asfollows:

TABLE 10 Reduced Well Model Properties Thickness, K_(x), K_(z), PI, PILayer ft mD mD b/d/psi Fraction 1 50 5 2.19 0.3 0.07 2 1 1000 0 1.210.29 3 50 7 3.66 0.42 0.10 4 1 1000 0 1.21 0.29 5 100 8.35 1.79 1.010.24 Calculated Bottom hole Potential Φ_(W) = 1421.34 psi

The reduced model in accordance with the present invention the proceedsto determine bottom hole potential Φ_(w). The results of the reducedmodel with five layers are set forth below in Table 11.

TABLE 11 Potential Distribution Layer Pot wf Pot 1 1421.34 2900.53 21421.34 1421.35 3 1421.34 2864.16 4 1421.34 1421.35 5 1421.34 2740.61

Using the bottom hole Potential Φ_(w) calculated from the reduced modeland computing potentials using specified bottom hole pressure Φ_(W) forthe full model, completion layer rates are calculated according toEquation (17). The results are indicated below in Table 12.

TABLE 12 Calculated Well Layer Rates New Fully Layer Method CoupledMethod  1 35.92 35.93  2 89.78 89.78  3 53.85 53.48  4 179.47 179.47  589.73 89.73  6 0.00 0.00  7 105.09 105.09  8 52.54 52.54  9 157.59157.60 10 210.09 210.10 11 87.55 87.55 12 0.00 0.00 13 129.28 129.29 14129.28 129.28 15 129.27 129.28 16 129.25 129.26 17 129.24 129.24 18158.48 158.49 158.49 19 158.41 158.42 20 158.37 158.37 21 158.33 158.34158.34 22 158.32 158.33 Newly calculated q_(t) = 2499.84 b/d Error =2,500. −2499.8488 = 0.15 b/d

Error in the total rate and computed bottom hole pressure vanishes withthe simulator's non-linear Newton iterations. The present inventionobtains a reduced model with a production rate acceptably accurate incomparison to the results obtained by the fully implicit, fully coupledprocessing techniques of the prior art, but with a substantial reductionin model complexity and computer processing time.

The present invention, as has been described above, does not require aspecial linear solver for the solution of the coupled reservoir and wellequations. In contrast, the coefficient matrix for the previously usedcoupled reservoir and well equations does not have regular sparsestructure. Therefore, the conventional types of coupled reservoir andwell equations require special solvers which can be expensive and canalso face convergence problems.

It can be seen that, as described above, the present invention does notrequire any special solver for the solution of coupled reservoir andwell equations. The same solver used for reservoir equations isutilized. The only modification made to the coefficient matrix is in thediagonal terms.

The present invention solves reservoir simulation problems where thevertical wells have many completions, which is a common occurrence inreservoirs. In recent simulation studies wells with more than 100vertical layers (completions) are very common. The fully coupled fullyimplicit well model with simultaneous solution is very expensive forthese cases. The present invention can save significant amounts ofcomputer time.

The present invention is very useful for wells having hundreds ofperforations completed in highly heterogeneous reservoirs. The presentinvention reduces the large, time consuming problem of well modelingsimulation in reservoirs with large numbers of layers (completions)problem to a small problem by recognizing and advantageously using thephysical principles involved. With the present invention, it has beenfound that vertically communicating layers can be lumped into a singlelayer. The reduced model so formed preserves the same bottom holepressure as the original full model. Once the reduced model is solvedfor the bottom hole pressure, the wells in the large system are thentreated as specified bottom hole pressure and solved easily by aconventional linear solver. Thus, the present invention eliminates theneed for writing or acquiring unstructured linear solvers for many wellswith hundreds of completions, which would be expensive.

The invention has been sufficiently described so that a person withaverage knowledge in the matter may reproduce and obtain the resultsmentioned in the invention herein Nonetheless, any skilled person in thefield of technique, subject of the invention herein, may carry outmodifications not described in the request herein, to apply thesemodifications to a determined structure, or in the manufacturing processof the same, requires the claimed matter in the following claims; suchstructures shall be covered within the scope of the invention.

It should be noted and understood that there can be improvements andmodifications made of the present invention described in detail abovewithout departing from the spirit or scope of the invention as set forthin the accompanying claims.

1. A computer implemented method of forming a model of determined layer completion rates of component fluids of perforated well intervals in a well in a subsurface reservoir, from measured total well production, with a coupled well reservoir model during reservoir simulation of well production at a time step during life of the subsurface reservoir, the coupled well reservoir model being organized into a reservoir grid subdivided into a plurality of reservoir cells at the perforated well intervals, the perforated well intervals in the reservoir being located at a plurality of formation layers, the formation layers being Nz in number and having unknown well potentials and fluid completion rates for component fluids at the time step, and the formation layers comprising vertical fluid flow layers having vertical fluid flow therefrom and flow barrier layers with no vertical fluid flow therefrom, the formation layers further having a permeability, thickness and a layer potential, the coupled well reservoir model further having a plurality of well cells at locations of the well in formation layers of the reservoir, the computer implemented method determining layer completion rates for the component fluids from the formation layers of the well and well production rates of the component fluids from the well, the computer implemented method comprising the steps of: (a) forming a full computation matrix reservoir model of reservoir data of cells of the model, including the reservoir data for the reservoir cells at the perforated well intervals, the reservoir data including the permeability, thickness and a potential for the formation layers; (b) forming a reduced well model system matrix by assembling as single vertical flow layers in the matrix the data of the vertical fluid flow layers of vertical fluid flow layers having vertical fluid flow therebetween and being located between flow barrier layers in the reservoir model; (c) determining a bottomhole pressure for the well; (d) forming a coupled reservoir well model comprising the full computation matrix reservoir model and the reduced well model system matrix, treating the well as a bottomhole, pressure specified well having the determined bottomhole pressure, the coupled reservoir well model being in the form of a matrix: ${\begin{bmatrix} A_{RR} & A_{RW} \\ A_{WR} & A_{WW} \end{bmatrix}\begin{bmatrix} \overset{\rightarrow}{\Phi_{R}} \\ \overset{\rightarrow}{\Phi_{W}} \end{bmatrix}} = \begin{bmatrix} \overset{\rightarrow}{b_{R}} \\ \overset{\rightarrow}{b_{W}} \end{bmatrix}$  wherein A_(RR) is an (Nz×Nz) tridiagonal matrix of the reservoir data, A_(RW) is an (Nz×1) vector of the productivity indexes of the formation layers adjacent the perforated well intervals; A_(WR) is an (1×Nz) vector of the productivity indexes from the well to the formation layers of the reservoir; A_(WW) is a (1×1) scalar of the productivity index of the well; {right arrow over (Φ_(R))} is a vector of unknown reservoir potentials for the cells around the well; {right arrow over (Φ_(W))} is a vector of unknown well potentials in the wellbore of the well; {right arrow over (b_(R))} is a matrix of reservoir data constants for the reservoir cells around the well; and {right arrow over (b_(W))} is a matrix of the well data constants for the well; (e) solving the coupled reservoir well model for the fluid flows in the reservoir cells of the formation layers and the productivities and potentials of the reservoir cells at the time step for each of the formation layers; (f) solving the coupled reservoir well model for the productivity indexes of the well cells at the perforated well intervals of the reservoir at the time step; (g) determining layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well based on the determined productivity indexes of the reservoir and well cells at the perforated well intervals of the reservoir at the time step; (h) determining total well production rate for the well from the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well at the time step; and (i) forming a record of the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well and the determined total well production rate for the well at the time step.
 2. The computer implemented method of claim 1, wherein the steps of solving the coupled well reservoir model comprise applying a full matrix solver.
 3. The computer implemented method of claim 2, further including the steps of: reducing residuals from the step of applying a full matrix solver against specified tolerances; and if the tolerances are satisfied at the time step, and for each time step during life of the subsurface reservoir, proceeding to the step of forming a record; and if the tolerances are not satisfied at the time step, returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the time step.
 4. The computer implemented method of claim 2, further including the steps of: reducing residuals from the step of applying a full matrix solver against specified tolerances; and if the specified tolerances are satisfied at the time step, but not for each time step during life of the reservoir, incrementing the simulator time step; and returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the incremented simulator time step.
 5. The computer implemented method of claim 1, wherein the step of solving the coupled well reservoir model comprises the step of solving for fluid flows based on the permeability, thickness and layer potentials of the formation layers.
 6. The computer implemented method of claim 1, wherein the well comprises a vertical well.
 7. The computer implemented method of claim 1, wherein the model is formed of determined layer completion rates of component fluids in a plurality of wells the subsurface reservoir.
 8. A data processing system forming a model of determined layer completion rates of perforated well intervals in a well in a subsurface reservoir, from measured total well production, with a coupled well reservoir model during reservoir simulation of well production at a time step during life of the subsurface reservoir, the coupled well reservoir model being organized into a reservoir grid subdivided into a plurality of reservoir cells at the perforated well intervals, the perforated well intervals in the reservoir being located at a plurality of formation layers, the formation layers being Nz in number and having unknown well potentials and fluid completion rates for component fluids at the time step, and the formation layers comprising vertical fluid flow layers having vertical fluid flow therefrom and flow barrier layers with no vertical fluid flow therefrom, the formation layers further having a permeability, thickness and a layer potential, the coupled well reservoir model further having a plurality of well cells at locations of the well in formation layers of the reservoir, the data processing system determining layer completion rates for the component fluids from the formation layers of the well and well production rates of the component fluids from the well, and comprising: a processor performing the steps of: (a) forming a full computation matrix reservoir model of reservoir data of cells of the model, including the reservoir data for the reservoir cells at the perforated well intervals, the reservoir data including the permeability, thickness and a potential for the formation layers; (b) forming a reduced well model system matrix by assembling as single vertical flow layers in the matrix the data of the vertical fluid flow layers of vertical fluid flow layers having vertical fluid flow therebetween and being located between flow barrier layers in the reservoir model; (c) determining a bottomhole pressure for the well; (d) forming a coupled reservoir well model comprising the full computation matrix reservoir model and the reduced well model system matrix, treating the well as a bottomhole, pressure specified well having the determined bottomhole pressure, the coupled reservoir well model being in the form of a matrix: ${\begin{bmatrix} A_{RR} & A_{RW} \\ A_{WR} & A_{WW} \end{bmatrix}\begin{bmatrix} \overset{\rightarrow}{\Phi_{R}} \\ \overset{\rightarrow}{\Phi_{W}} \end{bmatrix}} = \begin{bmatrix} \overset{\rightarrow}{b_{R}} \\ \overset{\rightarrow}{b_{W}} \end{bmatrix}$  wherein A_(RR) is a tridiagonal matrix of the reservoir data, A_(RW) is an (Nz×1) vector of the productivity indexes of the formation layers adjacent the perforated well intervals; A_(WR) is an (1×Nz) of the productivity indexes from the well to the formation layers of the reservoir; A_(WW) is a (1×1) scalar of the productivity index of the well; {right arrow over (Φ_(R))} is a vector of unknown reservoir potentials for the cells around the well; {right arrow over (Φ_(W))} is a vector of unknown well potentials in the wellbore of the well; {right arrow over (b_(R))} is a matrix of reservoir data constants for the reservoir cells around the well; and {right arrow over (b_(W))} is a matrix of the well data constants for the well; (e) solving the coupled reservoir well model for the fluid flows in the reservoir cells of the formation layers and the productivities and potentials of the reservoir cells at the time step for each of the formation layers; (f) solving the coupled reservoir well model for the productivity indexes of the well cells at the perforated well intervals of the reservoir at the time step; (g) determining layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well based on the determined productivity indexes of the reservoir and well cells at the perforated well intervals of the reservoir at the time step; (h) determining total well production rate for the well from the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well at the time step; and a memory performing the step of forming a record of the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well and the determined total well production rate for the well at the time step.
 9. The data processing system of claim 8, wherein the processor in solving the coupled well reservoir model performs the step of applying a full matrix solver.
 10. The data processing system of claim 9, further including the processor performing the steps of: reducing residuals from the step of applying a full matrix solver against specified tolerances; and if the tolerances are satisfied at the time step, and for each time step during life of the subsurface reservoir, causing the memory to perform the step of forming a record; and if the tolerances are not satisfied at the time step, returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the time step.
 11. The data processing system of claim 9, further including the processor performing the steps of: reducing residuals from the step of applying a full matrix solver against specified tolerances; and if the specified tolerances are satisfied at the time step, but not for each time step during life of the reservoir, incrementing the simulator time step; and returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the incremented simulator time step.
 12. The data processing system of claim 8, wherein the processor in solving the coupled well reservoir model performs the step of solving for fluid flows based on the permeability, thickness and layer potentials of the formation layers.
 13. The data processing system of claim 8, wherein the well comprises a vertical well.
 14. The data processing system of claim 8, wherein the model is formed of determined layer completion rates of component fluids in a plurality of wells the subsurface reservoir.
 15. A data storage device having stored in a non-transitory computer readable medium computer operable instructions for causing a data processor to form a model of determined layer completion rates of perforated well intervals in a well in a subsurface reservoir, from measured total well production, with a coupled well reservoir model during reservoir simulation of well production at a time step during life of the subsurface reservoir, the coupled well reservoir model being organized into a reservoir grid subdivided into a plurality of reservoir cells at the perforated well intervals, the perforated well intervals in the reservoir being located at a plurality of formation layers, the formation layers being Nz in number and having unknown well potentials and fluid completion rates for component fluids at the time step, and the formation layers comprising vertical fluid flow layers having vertical fluid flow therefrom and flow barrier layers with no vertical fluid flow therefrom, the formation layers further having a permeability, thickness and a layer potential, the coupled well reservoir model further having a plurality of well cells at locations of the well in formation layers of the reservoir, the stored computer operable instructions causing the data processor to determine layer completion rates for the component fluids from the formation layers of the well and well production rates of the component fluids from the well by performing the steps of: (a) forming a full computation matrix reservoir model of reservoir data of cells of the model, including the reservoir data for the reservoir cells at the perforated well intervals, the reservoir data including the permeability, thickness and a potential for the formation layers; (b) forming a reduced well model system matrix by assembling as single vertical flow layers in the matrix the data of the vertical fluid flow layers of vertical fluid flow layers having vertical fluid flow therebetween and being located between flow barrier layers in the reservoir model; (c) determining a bottomhole pressure for the well; (d) forming a coupled reservoir well model comprising the full computation matrix reservoir model and the reduced well model system matrix, treating the well as a bottomhole, pressure specified well having the determined bottomhole pressure, the coupled reservoir well model being in the form of a matrix: ${\begin{bmatrix} A_{RR} & A_{RW} \\ A_{WR} & A_{WW} \end{bmatrix}\begin{bmatrix} \overset{\rightarrow}{\Phi_{R}} \\ \overset{\rightarrow}{\Phi_{W}} \end{bmatrix}} = \begin{bmatrix} \overset{\rightarrow}{b_{R}} \\ \overset{\rightarrow}{b_{W}} \end{bmatrix}$  wherein A_(RR) is a tridiagonal matrix of the reservoir data, A_(RW) is an (Nz×1) vector of the productivity indexes of the formation layers adjacent the perforated well intervals; A_(WR) is an (1×Nz) vector of the productivity indexes from well to the formation layers of the reservoir; A_(WW) is a (1×1) scalar of the productivity index of the well; {right arrow over (Φ_(R))} is a vector of unknown reservoir potentials for the cells around the well; {right arrow over (Φ_(W))} is a vector of unknown well potentials in the wellbore of the well; {right arrow over (b_(R))} is a matrix of reservoir data constants for the reservoir cells around the well; and {right arrow over (b_(W))} is a matrix of the well data constants for the well; (e) solving the coupled reservoir well model for the fluid flows in the reservoir cells of the formation layers and the productivities and potentials of the reservoir cells at the time step for each of the formation layers; (f) solving the coupled reservoir well model for the productivity indexes of the well cells at the perforated well intervals of the reservoir at the time step; (g) determining layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well based on the determined productivity indexes of the reservoir and well cells at the perforated well intervals of the reservoir at the time step; (h) determining total well production rate for the well from the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well at the time step; and (i) forming a record of the determined layer completion rates for the component fluids of the vertical fluid flow layers and the flow barrier layers of the well and the determined total well production rate for the well at the time step.
 16. The data storage device of claim 15, wherein the instructions for causing the processor to perform the steps of solving the coupled well reservoir model comprise instructions for applying a full matrix solver.
 17. The data storage device of claim 15, wherein the instructions further include instructions causing the processor to perform the steps of: reducing residuals from the step of applying a full matrix solver against specified tolerances; if the tolerances are satisfied at the time step, and for each time step during life of the reservoir, proceeding to the step of forming a record; and if the tolerances are not satisfied at the time step, returning to step (a) and repeating steps (b) through (h) for another iteration of processing. at the time step.
 18. The data storage device of claim 15, wherein the instructions further include instructions causing the processor to perform the steps of: reducing residuals from the step of applying a full matrix solver against specified tolerances; and if the specified tolerances are satisfied at the time step, but not for each time step during life of the reservoir, incrementing the simulator time step; and returning to step (a) and repeating steps (b) through (h) for another iteration of processing at the incremented simulator time step.
 19. The data storage device of claim 15, wherein the instructions for causing the processor to perform the step of solving the coupled well reservoir model comprise instructions for solving for fluid flows based on the permeability, thickness and layer potentials of the formation layers.
 20. The data storage device of claim 15, wherein the well comprises a vertical well.
 21. The data storage device of claim 15, wherein model is formed of determined layer completion rates of component fluids in a plurality of wells the subsurface reservoir. 